## Introduction to Martingale Problems

**Abstract :**
The generator for a Markov process is a linear operator that
characterizes infinitesimally the evolution of the distribution of the
process. Classically, the Hille-Yosida theory of operator semigroups was
used to connect the Markov process to its generator. In the context of
diffusion processes, Stroock and Varadhan showed that Markov processes can
be characterized by the requirement that certain functionals of the process,
determined by the generator, must be martingales, that is, the Markov
process can be characterized as the unique solution of a **martingale
problem**. For example, Brownian motion is the unique solution of the
martingale problem for the Laplacian.

Martingale problems provide a powerful approach to the study of Markov
processes. The basic theory of martingale problems will be reviewed and
several illustrative examples of its application will be given.

NOTE: This talk is intended to give some useful background for the
November 12 talk in the Complex Systems Seminar.